I’m wondering if there is a way to compute the distance between nodes at each level of the nested SBM based on the consensus estimate obtained?

For example, consider that the inferred structure has 3 levels with 10, 2 and 1 blocks. I’m interested in the first level with 10 blocks. Can I compute how far node i and node j are based on this block structure?
Or would I only be able to infer whether node i and j belong to the same block?

Apologies if this has already been answered in this forum or on the website - I could not find any relevant information!

Ni! It really depends on what you want to describe.

For example, do you mean the distance between level-1 block nodes in the level-1 block graph? That is an awkward measure, which could be expressed as the smallest distance between their respective elements if you pretend that any two elements in the same block are connected with travel cost 0. Why would that be meaningful to you? Note that, in particular, A–B–C at level-1 doesn’t even imply that a path exists from an element in A to an element in C.

I would assume what one would more likely want to compute is the mean/median distance between the nodes belonging to each of two level-1 blocks. Which would translate as the typical distance to travel if you’re in a node in block A and want to go to a node in block B.

I suggest you try to think more clearly about your problem and what are you trying to describe.

Thank you for replying. I should have provided more details of what I’m interested in.
In my network I have a focal node of interest (say X), some nodes which interact in a specific manner with X (say nodes A, B and C) and all the remaining nodes in the network (D, E, F, G, …). This interaction is not directly reflected in the network structure as the network is based on spatial proximity while the interaction is a specific type of behaviour (I work with animal tracking data).

I’m interested in quantifying the ‘similarity’ of nodes (A, B, C) with X compared to the other nodes with X. My idea was that I could use the hierarchical network structure to show this similarity. Currently I use a simple metric of the proportion of specific nodes which are in the same block (so if A, B but not C are in the same block as X in level 1, the proportion is 2/3) and compare this to the proportion of other nodes which are in the same block as X.
However this seems like a very ‘crude’ measure and I’m wondering if there is some more appropriate/elegant metric that I can use.

Please let me know if I have completely misunderstood what the block structure implies.